Jacka Lu 發問時間: 科學數學 · 1 0 年前

國外友人請我幫問數學題目- 4

4. Let A(n) and B(n) be strictly increasing sequences and assume for all n, B(n)≠0.

a) Show that if k≠0 then the sequence C(n) = k × A(n) is strictly monotone

b) Give an example to show that D(n) = A(n)/B(n) could be a strictly decreasing sequence.

1 個解答

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  • 1 0 年前
    最佳解答

    a)

    since A(n) is strictly increasing

    A(n)<A(n+1) for all n

    if k>0

    kA(n)<kA(n+1)

    C(n)<C(n+1) for all n

    if k<0

    kA(n)>kA(n+1)

    C(n)>C(n+1) for all n

    so C(n) is strictly monotone

    b)

    Let A(n)=n+1 and B(n)=n

    then D(n)=A(n)/B(n)=1+1/n

    D(n)=1+1/n>1+1/(n+1)=D(n+1)

    D(n) is strictly decreasing

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